HALF-INTEGRAL WEIGHT MODULAR FORMS AND REAL QUADRATIC p-RATIONAL FIELDS JILALI ASSIM(1), ZAKARIAE BOUAZZAOUI(2) Abstract. Using half-integral weight modular forms we give a criterion for the existence of real quadratic p-rational fields. For p = 5 we prove the existence of infinitely many real quadratic p-rational fields. 1. Introduction The Dedekind zeta function of an algebraic number field encodes a lot of arithmetic infor- mation of the field. For a number field F , let F denote the ring of its integers. For each ∗ O integer m, let ζF (m) denote the leading non-zero coefficient in the Taylor expansion of the Dedekind zeta function of F . Dirichlet’s class number formula reads: ∗ hF ζF (0) = .RF , (1) −wF where hF is the class number of F , wF is the number of roots of unity in F and RF is the Dirichlet regulator. We are interested with the divisibility, by odd prime numbers p, of the special values of Dedekind zeta functions of real quadratic fields at odd negative integers, these values are closely related to the orders of certain cohomology groups. Let S be a finite set of primes. Denote by FS the maximal pro-p-extension of F which is unramified outside S and let GS(F ) be its Galois group. The field F is called p-rational if the Galois group GSp (F ) of the extension FSp /F is pro-p-free (with rank 1 + r2, r2 being the number of complex primes of F ), where Sp is the set of primes of F above p. If F is totally real, we prove in section 2 that F is p-rational precisely when vp(ζF (2 p)) = 1, − − where vp denotes the p-adic valuation. We use this characterization to study p-rationality of real quadratic fields. The notion of p-rational fields has been introduced to construct extensions of Q satisfying the Leopoldt conjecture [M-N]. Recently, R. Greenberg [G] used p-rational number fields to construct (in a non geometric manner) Galois representations with open image in GLn(Zp) for n 3. This paper is motivated by the study of p-rationality ≥ arXiv:1906.03344v1 [math.NT] 7 Jun 2019 of multi-quadratic number fields, for which Greenberg formulated the following conjecture: Conjecture 1. ([G, Conjecture 4.2.1]) For any odd prime number p and any integer t 1, t ≥ there is a p-rational field F such that Gal(F/Q) ∼= (Z/2Z) . The conjecture is true for t = 1, since for every odd prime number p, there is infinitely many p-rational imaginary quadratic fields (cf. [G, Proposition 4.1.1]). The case t 2 leads to the study of p-rationality of real quadratic fields, which is the aim of this paper.≥ After relating the p-rationality to special values of L-functions, we use the theory of modular forms to obtain our results. Roughly speaking, we use Cohen-Eisenstein series [C], which are modular forms of half integer weight, and whose Fourier coefficients involve special values of L-functions of quadratic fields. Multiplying such modular forms by theta series produces integer weight modular forms, and the resulting Fourier coefficients are studied to deduce divisibility properties of values of L-functions. As a consequence we give for p = 5 the 2010 Mathematics Subject Classification. 11R11, 11F37, 11R42. Key words and phrases. L-functions, p-rational field, modular forms. 1 2 JILALIASSIM,ZAKARIAEBOUAZZAOUI existence of infinitely many real quadratic 5-rational fields, a similar result for p = 3 was given implicitly by D. Byeon in [By] using the same techniques. Theorem 1.1. There are infinitely many fundamental discriminants d> 0 such that Q(√d) is 5-rational. The study of p-rationality of real quadratic fields is more subtle than the study of p- rationality of imaginary quadratic fields, because of complications due to the existence of non- trivial units. Using Cohen-Eisenstein series, Theorem 1.2 below gives a sufficient condition for the existence of a real quadratic p-rational field, with some arithmetic properties, for every prime number p 5. More precisely, let f = a(n)qn be an integer weight modular ≥ n≥0 form for the congruence subgroup Γ(N), N 1,P with coefficients in the ring of integers of a number field. By a result of Serre [S76, page≥ 20-19], there is a set of primes ℓ 1 (mod Np2) of positive density for which ≡ f T (ℓ) 2f (mod p2), (2) | ≡ where T (ℓ) denotes the Hecke operator associated to the prime number ℓ [Ko, page.153]. Let = ℓ1, ..., ℓs be a finite set of odd primes. For every positive square free integer L { } 2 s 4 t, let f be an element of the space Mp(Γ1(4p t i=1 ℓi )), obtained by multiplication of half integer weight modular forms (Cohen-EisensteinQ series and theta series). Denote by t the set of primes ℓ satisfying (2) for f. We make the following hypothesis: S 2 2 (Hp): There exist a square free integer t and a prime number ℓ t such that ℓ = ta + b and b is a prime number for which p is non-Wieferich. ∈ S Theorem 1.2. Let = ℓ , ..., ℓs be a finite set of odd primes. Let p 5 be a prime L { 1 } ≥ number. Assume that hypotheses (Hp) is satisfied for some prime number ℓ. Then there is a real quadratic p-rational field Q(√d) for some fundamental discriminant d<ℓ such that d . ( )=1 for every ℓk , where ( ) denotes the Legendre symbol. ℓk ∈L ℓk 2. p-rationality of quadratic fields ′ 1 Let p be an odd prime number and let F = F [ p ] be the ring of p-integers of F , then O O 2 ′ the field F is called p-rational if the étale cohomology group H ( F , Z/pZ) vanishies [M-N], 2 ′ O [Mo88], [Mo90]. In general, for every integer i, if H ( F , Z/pZ(i)) = 0 then we say that the field F is (p, i)-regular [A]. If F is totally real, theO information about the p-rationality and the (p, i)-regularity of F are contained in special values of the Dedekind zeta function ζF at odd negative integers. More precisely, as a consequence of the Main Conjecture in Iwasawa theory for totally real number fields and odd primes p proved by A.Wiles, we obtain the following case of Lichtenbaum conjecture: for any even positive integer i 2, and any totally real number field F , we have ≥ 2 ′ wi(F )ζF (1 i) p H ( , Zp(i)) , (3) − ∼ | OF | 0 where wi(F ) is the order of the group H (F, Qp/Zp(i)) and p means that they have the same p-adic valuation. Moreover, a periodicity statement on coho∼ mology groups gives that H2( ′ , Z/pZ(i)) = H2( ′ , Z/pZ(j)), OF ∼ OF whenever i j (mod p 1). Then we have the following proposition: ≡ − Proposition 2.1. Assume that i 2 is an even integer, then a totally real number field F ≥ is (p, i)-regular if and only if wi(F )ζF (i) is a p-adic unit. HALF-INTEGRAL WEIGHT MODULAR FORMS AND REAL QUADRATIC p-RATIONAL FIELDS 3 Proof. The proof follows from (3) and Proposition 2.4 of [A]. Suppose that F is a totally real number field of degree g. Let vp be the p-adic valuation. We have for even positive integers i the following result [S71, Theorem 6]: Theorem 2.2. Let p be an odd prime number. (1) if gi 0 (mod p 1), vp(ζF (1 i)) 1 vp(g); ≡ − − ≥ − − (2) if gi 0 (mod p 1), vp(ζF (1 i)) 0. 6≡ − − ≥ In particular, we have vp(ζF (2 p)) 1 vp(g). (4) − ≥ − − In [S, Section 3.7], it is suggested that often vp(ζF (2 p)) 1. Using Formula (3) we relate − ≤ − the p-rationality of F to the special value ζF (2 p) in the following way: Let p 3 be a prime which is unramified in F ,− then ≥ F is p-rational vp(ζF (2 p)) = 1. (5) ⇔ − − For F = Q(√d) a real quadratic fields, the Dedekind zeta function of F satisfies ζF (2 p)= ζQ(2 p)L(2 p, χd). − − − Since the field Q is p-rational for every odd prime number p (which is equivalent to say that vp(ζQ(2 p)) = 1), we have the following proposition: − − Proposition 2.3. Assume that p ∤ d, then the field Q(√d) is p-rational precisely when vp(L(2 p, χd))=0. − This is the motivation behind using the half-integer weight modular forms called Cohen- Eisenstein series described in the next section. 3. Cohen-Eisenstein series Let d< 0 be a fundamental discriminant and denote by h(d) the class number of Q(√d). For a rational prime p 5, it is known that if p ∤ h(d) then the field Q(√d) is p-rational. An object which generate≥ class numbers of imaginary quadratic fields is the 3-power of the standard theta series θ given by the q-expansion θ(q)=1+2q +2q4 +2q9 + ···· This series has been used to prove the existence of infinitely many p-rational imaginary 3 3 quadratic fields. More precisely, the series θ is a modular form of weight 2 for the congruence subgroup Γ0(4). Write 3 n θ (q)= r3(n)q , nX≥0 then the coefficient r3(n) is the number of times we can write n as a sum of three squares.